Can a linear mapping ever be surjective? I can't come up with an example.
Linear transformations are matrices, and a matrix is invertible if and only if it is square with non-zero determinant. That is, matrices with non-zero determinant are bijective linear transformations. So every invertible matrix corresponds to a surjective linear map.
(...for a suitable value of are...)
If you want, you could try and use matrices to classify all surjective linear maps. It is a good exercise, methinks.